So you dump the grains in a jar, add milk, and let it do its thing. To use the kefir, you remove the grains from the jar - they float on top, so I just pick them out with my fingers, keeping track of how many there are, transfer them to a clean jar and fill it with more milk. Theoretically. This worked great for me in the summer, but when I returned from vacation it seemed my grains were suffering from some unheard-of kefir illness, and they wouldn't work properly. It turns out that they just didn't like the cooler weather - my kitchen was running at consistently about 25-28C over the summer and that's the temperature they like. I solved the problem by sticking the jar in the oven and leaving the oven light on, although I recommend some visual cue (like a piece of paper taped over the oven control) to remind you, otherwise you might cook your grains by mistake. I've had a few close calls!
When it's done, your kefir jar should look like this. This jar is slightly atypical in that the whey layer on top is not usually so pronounced - I shook the jar when it was a little far along and trapped the grains under the top layer of cheesyness, creating more coagulation at the top and thus more whey. But that's not important. The important thing to look for when deciding if your kefir "worked" is the little pockets of whey at the bottom of the jar. This means that the coagulation has reach the bottom of the jar and your kefir is nicely thickened. If you leave it longer, those pockets will get bigger and the coagulation will become more pronounced until you have a jar with just a cheesy layer on top half and clear whey on the bottom half. There's nothing wrong with this, it just tastes a little strong. It's still completely edible.